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Let R be a relation between A and B. R is asymmetric if and only if ________
- a)Intersection of D(A) and R is empty, where D(A) represents diagonal of set
- b)R-1 is a subset of R, where R-1 represents inverse of R
- c)Intersection of R and R-1 is D(A)
- d)D(A) is a subset of R, where D(A) represents diagonal of set
Correct answer is option 'A'. Can you explain this answer?
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Let R be a relation between A and B. R is asymmetric if and only if __...
A relation is asymmetric if and only if it is both antisymmetric and irreflexive. As a consequence, a relation is transitive and asymmetric if and only if it is a strict partial order. If D(A) is a diagonal of A set and intersection of D(A) and R is empty, then R is asymmetric.
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Let R be a relation between A and B. R is asymmetric if and only if __...
Asymmetric Relation and its Characteristics:
An asymmetric relation between sets A and B is a relation that does not have any pairs where both elements are related to each other. In other words, if (a, b) is in relation R, then (b, a) cannot be in R.
To understand why option 'A' is the correct answer, let's examine the characteristics of an asymmetric relation:
1. No symmetric pairs: In an asymmetric relation, there are no pairs where both elements are related to each other. This means that if (a, b) is in R, then (b, a) cannot be in R.
Explanation of Option 'A':
Option 'A' states that the intersection of the diagonal of set A (D(A)) and relation R is empty. Let's break down this statement to understand why it is true for an asymmetric relation:
- Diagonal of set A (D(A)): The diagonal of set A, denoted as D(A), is a set of ordered pairs where both elements are the same. For example, if A = {1, 2, 3}, then D(A) = {(1, 1), (2, 2), (3, 3)}. It represents the pairs where an element is related to itself.
- Intersection of D(A) and R: The intersection of D(A) and R represents the pairs where both elements are related to each other in the relation R.
- Empty intersection for asymmetric relation: In an asymmetric relation, there are no symmetric pairs, i.e., no pairs where both elements are related to each other. Therefore, the intersection of D(A) and R will be empty for an asymmetric relation.
Hence, option 'A' is the correct answer because an asymmetric relation will have an empty intersection between the diagonal of set A and relation R.
Other Options Explanation:
- Option 'B': R-1 represents the inverse of relation R. If R is asymmetric, it means that (a, b) is in R implies (b, a) is not in R. However, this does not imply that the inverse of R, R-1, is a subset of R.
- Option 'C': The intersection of R and R-1 represents the pairs that are both in R and its inverse R-1. This intersection does not necessarily correspond to the diagonal of set A.
- Option 'D': D(A) represents the diagonal of set A, and it consists of pairs where an element is related to itself. However, an asymmetric relation does not require all elements of set A to be related to themselves, so D(A) being a subset of R is not a characteristic of an asymmetric relation.
An asymmetric relation between sets A and B is a relation that does not have any pairs where both elements are related to each other. In other words, if (a, b) is in relation R, then (b, a) cannot be in R.
To understand why option 'A' is the correct answer, let's examine the characteristics of an asymmetric relation:
1. No symmetric pairs: In an asymmetric relation, there are no pairs where both elements are related to each other. This means that if (a, b) is in R, then (b, a) cannot be in R.
Explanation of Option 'A':
Option 'A' states that the intersection of the diagonal of set A (D(A)) and relation R is empty. Let's break down this statement to understand why it is true for an asymmetric relation:
- Diagonal of set A (D(A)): The diagonal of set A, denoted as D(A), is a set of ordered pairs where both elements are the same. For example, if A = {1, 2, 3}, then D(A) = {(1, 1), (2, 2), (3, 3)}. It represents the pairs where an element is related to itself.
- Intersection of D(A) and R: The intersection of D(A) and R represents the pairs where both elements are related to each other in the relation R.
- Empty intersection for asymmetric relation: In an asymmetric relation, there are no symmetric pairs, i.e., no pairs where both elements are related to each other. Therefore, the intersection of D(A) and R will be empty for an asymmetric relation.
Hence, option 'A' is the correct answer because an asymmetric relation will have an empty intersection between the diagonal of set A and relation R.
Other Options Explanation:
- Option 'B': R-1 represents the inverse of relation R. If R is asymmetric, it means that (a, b) is in R implies (b, a) is not in R. However, this does not imply that the inverse of R, R-1, is a subset of R.
- Option 'C': The intersection of R and R-1 represents the pairs that are both in R and its inverse R-1. This intersection does not necessarily correspond to the diagonal of set A.
- Option 'D': D(A) represents the diagonal of set A, and it consists of pairs where an element is related to itself. However, an asymmetric relation does not require all elements of set A to be related to themselves, so D(A) being a subset of R is not a characteristic of an asymmetric relation.
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Let R be a relation between A and B. R is asymmetric if and only if ________a)Intersection of D(A) and R is empty, where D(A) represents diagonal of setb)R-1is a subset of R, where R-1represents inverse of Rc)Intersection of R and R-1is D(A)d)D(A) is a subset of R, where D(A) represents diagonal of setCorrect answer is option 'A'. Can you explain this answer?
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Let R be a relation between A and B. R is asymmetric if and only if ________a)Intersection of D(A) and R is empty, where D(A) represents diagonal of setb)R-1is a subset of R, where R-1represents inverse of Rc)Intersection of R and R-1is D(A)d)D(A) is a subset of R, where D(A) represents diagonal of setCorrect answer is option 'A'. Can you explain this answer? for Civil Engineering (CE) 2025 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Let R be a relation between A and B. R is asymmetric if and only if ________a)Intersection of D(A) and R is empty, where D(A) represents diagonal of setb)R-1is a subset of R, where R-1represents inverse of Rc)Intersection of R and R-1is D(A)d)D(A) is a subset of R, where D(A) represents diagonal of setCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let R be a relation between A and B. R is asymmetric if and only if ________a)Intersection of D(A) and R is empty, where D(A) represents diagonal of setb)R-1is a subset of R, where R-1represents inverse of Rc)Intersection of R and R-1is D(A)d)D(A) is a subset of R, where D(A) represents diagonal of setCorrect answer is option 'A'. Can you explain this answer?.
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